University of Pennsylvania

ScholarlyCommons Departmental Papers (MEAM)

Department of Mechanical Engineering & Applied Mechanics

5-1-2005

Analysis of sedimentation biodetectors Shizhi Qian University of Pennsylvania

Raimund Bürger University of Stuttgart Pfaffenwaldring

Haim H. Bau University of Pennsylvania, [email protected]

Postprint version. Published in Chemical Engineering Science, Volume 60, Issue 10, May 2005, pages 2585-2598. Publisher URL: http://dx.doi.org/10.1016/j.ces.2004.12.014 This paper is posted at ScholarlyCommons. http://repository.upenn.edu/meam_papers/129 For more information, please contact [email protected].

Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

Analysis of Sedimentation Biodetectors Shizhi Qiana, Raimund Bürgerb, Haim H. Baua+ a

Mechanical Engineering and Applied Mechanics, University of Pennsylvania Philadelphia, PA 19104, USA

b

Institute of Applied Analysis and Numerical Simulation, University of Stuttgart Pfaffenwaldring 57, D-70569 Stuttgart, Germany

ABSTRACT A bead-based sedimentation biodetector is studied theoretically. The biodetector operates with a suspension of settling beads, non-settling reporters, and target analytes – all initially suspended in a buffer solution. The reporters can be either fluorescent molecules or small particles. The functionalized beads interact with the reporters and target analytes while settling under the action of gravitational, electric, and/or magnetic fields. Both sandwich and competitive assays with hindered settling are considered. In the sandwich format, in the presence of target analytes, the reporters bind to the beads and settle (the target analytes provide the link between the beads and the reporters). A reduction in the reporters’ concentration indicates the presence of target analytes. In the competitive format, both target analytes and reporters compete for bead-based binding sites. In the absence of target analytes, one would observe a reduction in the suspended reporters’ concentration. The model allows one to predict the reporters’ concentration in solution as a function of initial bead, reporter, and target analyte concentrations and provides a means for the reactor’s optimization. Keywords: Sedimentation, Biodetector, Competitive assay, Sandwich assay

1. Introduction +

Corresponding author E-mail addresses: [email protected] (S. Qian), [email protected] (R. Bürger),

[email protected] (H. H. Bau)

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

In recent years, there has been a growing interest in developing low-cost techniques for inexpensive, rapid identification of various pathogens at the point of care. For example, the lateral flow immunoassay is a popular diagnostic tool because it eliminates the need for trained personnel and expensive equipment and provides rapid diagnostics at the point of care (Qian and Bau, 2003, 2004). The lateral flow immunoassay consists of a nitrocellulose membrane in which reporter particles and target analytes are propelled to an interaction zone by capillary forces. Unfortunately, relatively large membrane-to-membrane variations, the adhesion of reporter particles and target analytes to the membrane, and the presence of significant background noise reduce the sensitivity of this format. An interesting alternative is the sedimentation reactor (Lim, 1990; Lim and Ko, 1990; Lim et al., 1998; House et al., 2001; Oracz et al., 2003; Tam et al., 2003; Feleszko et al., 2004). The sedimentation assay consists of functionalized beads (B), functionalized reporter particles (P), and target analytes (A). The beads are typically much larger and tend to settle much faster than the reporter particles. The latter can stay in solution for a very long time. The settling process can be accelerated with the use of centrifugal forces or magnetic fields (when the beads are made of a magnetic material). The reporter particles may consist of colored particles, fluorescent labels, magnetic materials, or phosphor particles. The detection technique is dictated by the nature of the reporter particles. For example, colored particles can be detected visually while phosphor particles are typically excited with a laser and their emission is measured with a photo-detector. Two different assays are possible: sandwich and competitive. In the sandwich format, the target analyte (A) binds to both the beads (B) and the reporter particles (P) to form the complexes BA and AP. The complex BA can bind with P or the complex AP can bind with B to form the sandwich complex BAP. The beads and their complexes settle to the bottom of the reactor while the free target analytes and reporter particles remain in solution. Figs. 1a and b sketch, respectively, the processes in the absence and presence of target analytes. In the absence of or at low concentrations of target analyte (A), the reporter CES-D-04-00047

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

particles (P) cannot bind to the heavier settling beads (B). They remain suspended, and there is no change in the supernatant’s color or signal intensity. In the presence of target analytes, some of the reporter particles bind to the beads and settle. This leads to a reduction in the supernatant’s signal. As the target analyte concentration increases, the supernatant’s signal intensity decreases. In the competitive format, the target analyte (A) and the functionalized reporter particles (P) can competitively bind to the functionalized heavier beads (B) as they settle to the bottom of the reactor. When the analyte (A) is absent, most of the reporter particles bind to the beads, and there is an obvious color change in the supernatant. When there is an abundance of target analytes, the target molecules occupy many of the binding sites on the beads, and most of the reporter particles remain in solution. Hence, little or no change in the supernatant’s color indicates the presence of an abundance of target analytes. Figs. 2a and 2b depict the competitive process in the absence and the presence of target analytes, respectively. The TUBEXTM (IDL Biotech, Sollentuna, Sweden) used to detect anti-O9 (immunoglobulin M (IgM) mouse hybridoma) antibodies is an example of a sedimentation reactor operating with a competitive assay (Lim, 1990; Lim and Ko, 1990; Lim et al., 1998; House et al., 2001; Oracz et al., 2003; Tam et al., 2003; Feleszko et al., 2004). In this immunoassay, colored latex particles coated with anti-O9 mAb and magnetic particles coated with Salmonella typhi LPS are mixed in a tube with the sample to be examined. Subsequently, the reactor tube is placed on a magnet, and the magnetic beads settle to the bottom of the tube. The detection results are based on the concentration of the indicator particles that remain suspended as indicated by the color of the supernatant. The designers of bioassays typically employ empirical means to optimize the assay format (i.e., the selection of the optimal bead and reporter particle concentrations needed to achieve high sensitivity at pre-specified target analyte concentrations). It appears that it would be desirable to have a predictive tool that can provide quantitative information. To the best of our knowledge, such a mathematical model accounting for the effects of hindered settling has CES-D-04-00047

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

not been developed. This paper takes a few, first steps in the development of such a modeling tool. Sedimentation of polydisperse suspensions of solid particles with different sizes and densities are widely used in unit operations; materials, minerals, food, and pharmaceuticals processing; and wastewater treatment (Sharma et al., 1993; Concha and Bürger, 2002; Berres et al., 2003; Xue and Sun, 2003). Bürger and Wendland (2001) and Concha and Bürger (2002) review sedimentation research with a focus on mineral processing. Several mathematical models, based on multiphase flow theory for the sedimentation of monodisperse or polydisperse suspensions, with and without considering sediment compressibility, have also been proposed (Smith, 1965, 1966; Lockett and Al-Habbooby, 1973; Mirza and Richardson, 1979; Masliyah, 1979; Lockett and Bassoon, 1979; Batchelor, 1982; Batchelor and Wen, 1982; Selim et al., 1983; Shih et al., 1987; Davis and Gecol, 1994; Bürger and Tory, 2000; Bürger et al., 2000a, 2000b, 2000c, 2001, 2002; Xue and Sun, 2003; Berres et al., 2003, 2004a, 2004b; Tory and Ford, 2004). The predictions of the mathematical models, solved numerically by recently developed finite difference schemes for conservation laws under various batch and continuous flow conditions, favorably agree with experiments (Berres et al., 2003, 2004a, 2004b, 2005; Xue and Sun, 2003; Bürger et al., 2000c; Bürger et al., 2001; Zeidan et al., 2004). A simpler Lamm equation has also been used in the analysis of centrifugal sedimentation reactors (Schuck, 1998, 2004a, 2004b, and the references cited therein, and Stafford and Braswell, 2004). In fact, the Lamm equation is a simpler variant of the multiphase flow models, and it neglects the effect of hindered settling and includes diffusion terms that ensure that the solutions of the equations are smooth. However, hindered settling is important, especially when the solid volume fraction is great enough to inhibit liquid movement and liquid must move in the spaces between particles. Yet none of the existing models accounts for chemical reactions and biological interactions with hindered settling that may occur during the sedimentation process of polydisperse suspension. The

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objective of this paper is to propose such a model, which would be useful for the design and optimization of sedimentation biodetectors working with competitive and sandwich assays. The paper is organized in the following way. Sections 2 and 3 extend, respectively, the Masliyah-Lockett-Bassoon (MLB) model (Masliyah, 1979; Lockett and Bassoon, 1979) as described by Bürger et al (2002) and the high-resolution Kurganov-Tadmor central-difference scheme (Kurganov and Tadmor, 2000) to account for biological interactions. The presence of shocks requires the use of a shock-capturing scheme. Section 4 provides a few examples of the calculations, and section 5 concludes. Unfortunately, we were not able to find any quantitative experimental data to compare with our calculations.

2. Mathematical Model Consider an upright cylindrical sedimentation reactor of height L, initially filled with a homogeneous suspension of Ns species of protein-conjugated particles, (Nf –1) target analytes, and a buffer solution. In total, the solution consists of N=Ns+Nf species. The target analytes are assumed to be present at very low concentrations, to have a negligible effect on the buffer solution’s density and viscosity, and to translate at the velocity of the buffer. In contrast, the particle species have a significant effect on the solution’s properties and move at velocities different than the surrounding buffer. We describe the suspension as a superposition of continua (Drew and Passman, 1999). Both the liquid and solid media are treated as viscous fluids. The model presented here is an extension of the treatment given in Bürger et al (2002).

2.1 Mass and Linear Momentum Balance Equations In this section, we formulate the mass and momentum conservation equations. We consider the target analytes and the buffer to be a single phase. By definition, the volume fractions φ j

( j = 1,..., N s )

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of all the solid phases and the liquid sum up to one:

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598 N s +1

∑φ j =1

j

= 1,

(1)

where φ = φ1 + K + φ N s is the total volume fraction occupied by all solid particle species and

φN

s +1

(

= 1 − φ is the volume fraction of the fluid phase. We define a vector Φ = φ1 , K, φ N s

) for T

later use. Hereafter, bold letters denote vectors. The continuity equation for each solid phase is

∂ρ iφi + ∇ ⋅ ( ρ iφi v i ) = mis , ∂t

i = 1,..., N s

(2)

where v i is the ith solid phase velocity vector, mis is the rate of production of the ith solid phase, and ρ i is the (constant) density of the ith solid phase. Both the liquid and solid phases are incompressible. The first term on the left hand side of equation (2) accounts for the rate of mass accumulation per unit volume, and the second term is the net rate of convective mass flux. The term on the right accounts for the interphase mass transfer resulting from biological interactions. We neglect mass transport due to diffusion. The continuity equation of each species in the fluid phase is:

∂ρ f φ N +1Yi + ∇ ⋅ ( ρ f φ N +1Yi v f ) = mif , i = 1,..., N f ∂t s

(3)

s

where v f is the fluid phase’s velocity; ρ f is the density of the fluid phase; and mif and Yi are, respectively, the rate of production and the mass fraction of the ith species in the fluid phase. By definition, Nf

∑Y i =1

i

= 1.

We define the vector Χ = ( X 1 , K , X N f ) , where Χ i = φ N s +1Yi = (1 − φ )Yi and T

(4) Nf

∑X i =1

i

= (1 − φ ) .

Equation (3) can be expressed in terms of Xi as:

∂X i mif + ∇⋅(Xi v f ) = , i = 1,..., N f . ρf ∂t

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(5)

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

Conservation of mass requires that the net mass transfer over all phases must be zero, Nf

Ns

∑m + ∑m i =1

s i

i =1

f i

= 0.

(6)

Summing up the individual equations (5), we have

−

∂φ 1 + ∇ ⋅ [(1− φ)v f ] = ∂t ρf

Nf

∑m . f i

(7)

q := (1 − φ ) v f + ∑ φi v i .

(8)

i=1

The volume-average velocity of the suspension is: Ns

i =1

Dividing the i-th equation in (2) by ρ i , summing the resulting equations over i=1,…,Ns, adding the result to equation (7), and using the constraint (6), we obtain: Ns ⎛ 1 1 ⎞⎟ ∇ ⋅ q = ∑ mis ⎜ − . ⎜ρ ⎟ i =1 ⎝ i ρf ⎠

(9)

The momentum equation for each solid phase is: N V ⎡ ∂vi ⎤ + ( vi ⋅ ∇) vi ⎥ = −∇(φi p) + ∇ ⋅ T i + ρiφi b + I if + ∑I ik + mis vi , i = 1,...,N s k =1 ⎦ ⎣ ∂t k ≠i s

ρiφi ⎢

(10)

V

where p is the pressure; T i is the viscous part of the stress tensor of the ith particle species (the particle species are treated as pseudofluids); b is the body force density; I if is the interaction force representing the momentum transfer between the ith particle species and the fluid phase;

I ik is the interaction force between the ith and kth particle species; and mis v i describes the momentum transfer associated with the mass transfer. Similarly, the momentum equation for the fluid phase is: Nf Ns V ⎡ ∂v f ⎤ + ( v f ⋅ ∇) v f ⎥ = ∇((1 − φ ) p ) + ∇ ⋅ T f + ρ f (1 − φ )b − ∑ I if + ∑ mif v f . i =1 i =1 ⎣ ∂t ⎦

ρ f (1 − φ )⎢

(11)

The terms on the right hand side of equation (11) represent, respectively, the pressure, the viscous part of the fluid phase stress tensor, the body force, the interaction forces between the CES-D-04-00047

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

fluid phase and all solid phases, and the momentum transfer due to biological interactions of all the species in the fluid phase. Although the model can accommodate body forces resulting from magnetic, electrical, and centrifugal forces, we will consider here only the case of the gravitational force b = − gkˆ , where kˆ is the upward-pointing unit vector.

2.2 Interaction Forces The interaction force between the fluid and the ith solid species is modeled by

I if = α i (Φ)u i + p∇φi ,

i = 1,..., N s

(12)

where u i = v i − v f is the slip velocity of the particle species i, and α i (Φ) is the resistance coefficient (Bürger et al., 2002)

φi α i (Φ)

=−

d i2V (φ ) ; 18μ f

(13)

μ f is the viscosity of the fluid;

⎧(1 − φ ) n − 2 V (φ ) = ⎨ ⎩0

(n > 2 )

for 0 ≤ φ ≤ φ max , otherwise

(14)

is the hindered settling factor (Richardson and Zaki, 1954); and φmax is the volume fraction of the settled particles. The interactions among the different solid particle species could be specified by the Nakamura and Capes formula (Nakamura and Capes, 1976; Arastoopour et al., 1982; Shih et al., 1987; Bürger et al., 2002). Since these interaction forces can be neglected in our case (see section 2.4), we do not reproduce the explicit expressions here. Introducing relationship (12) into the momentum equations, we obtain, respectively, the modified momentum equations for the solid and fluid phases:

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598 Ns V ⎡ ∂v i ⎤ + (v i ⋅ ∇ )v i ⎥ = −φi ∇p + ∇ ⋅ T i + ρ iφi b + α i (Φ)u i + ∑ I ik + mis v i , i = 1,..., N s ⎣ ∂t ⎦ k =1

(15)

Ns Ns V ⎡ ∂v f ⎤ + ( v f ⋅ ∇) v f ⎥ = −(1 − φ )∇p + ∇ ⋅ T f + ρ f (1 − φ )b − ∑α i (Φ)u i − ∑ mis v f . i =1 i =1 ⎣ ∂t ⎦

(16)

ρ iφi ⎢

k ≠i

and

ρ f (1 − φ )⎢

2.3 Mass Transfer Due to Biological Interactions The rates of mass production, m1s ,K , m Ns s and m1f ,K , m Nf f , are the result of the biological interactions that occur during the sedimentation process.

Since the concentrations of

the particles and target analytes are very low, we assume reversible, 1:1 interactions: k ai

C a (i ) + Cb (i ) ⇔ Cc (i ) ,

i = 1,..., R

(17)

k di

In other words, the binding of multiple target analytes and/or reporter particles to a single bead is a low probability event. In the above, R is the total number of possible interactions; k ai and k di are, respectively, the association and dissociation rate constants of the ith interaction; and C a (i ) , C b (i ) and C c (i ) denote the various species involved in the ith interaction, each of which corresponds to one of the particles or fluid species. The rate of formation of the jth species is:

[C′j ] = ∑{kai (δ j,c(i ) − δ j,a(i ) − δ j,b(i ) ) [Ca(i ) ][Cb(i) ] − kdi (δ j,c(i) − δ j,a(i) − δ j ,b(i) )[Cc(i) ]}, j = 1,..., N R

(18)

i =1

where the square brackets [ ] denote molar concentration; X ′ = dX dt ; and δ i , j is the Kronecker delta ( δ i , j = 1 when i = j and δ i , j = 0 when i ≠ j ). The “molar” concentration of the solid particles is the ratio of the number of particles per liter divided by the Avogadro number. The molar concentration of the target molecules has its usual meaning [Ci ] = ρ f X i MWi ,

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

where MWi is the molecular mass of the ith species. When we consider particles, [Ci ] = ρ sφi MWi and MWi is the mass of individual particles. The rate of mass transfer vector, m = (m1 ,K , m N ) T = (m1s ,K , m Ns s , m1f ,K m Nf f ) T , where

mi (W) = MWi [Ci′],

i = 1,..., N

(19)

is a function of the vector W =( W1 ,...,W N s , W N s +1 ,K , W N ) T = (φ1 ,..., φ N s , X 1 , K , X N f ) T .

(20)

2.4 Order of Magnitude Estimates The momentum equations for the solid (15) and fluid (16) phases are quite complicated. Fortunately, order of magnitude analysis allows one to demonstrate that certain terms are unimportant and that the equations can be significantly simplified (Bürger et al., 2002; Berres et al., 2003). We use ρ f as the density scale; the velocity U of the fastest settling particle in an unbounded medium as the velocity scale; the height of the device L as the length scale; the settling time L/U as the time scale; and the hydrostatic pressure ρ f gL as the pressure scale. The representative kinematic viscosities of the solid and fluid phases are denoted, v0s and v0f , respectively. The dimensionless momentum equations for the solid phases and the liquid are, respectively,

ρ i*φi Fr

V d 1 v0s Fr * Dv *i * * * * * * ˆ p T = − ∇ + ∇ ⋅ ( φ i ) − ρ i φ i k + α i (Φ )u i + i * f Dt L v0 Re N L misU * k * Fr ∑ (I i ) + vi , d1 ρf g k =1 s

(21)

i = 1,..., N s

k ≠i

and

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

∇ * p * = − kˆ −

N Dv f 1 α i* (Φ )u *i − Fr ∑ Dt * (1 − φ ) i =1 . N V U 1 d 1 Fr * m is v *f + ∇ ⋅ (T f ) * − ∑ (1 − φ ) L Re ρ f g (1 − φ ) i =1 *

s

(22)

s

In the above, the superscript star denotes dimensionless quantities. The Froude number

Fr = U 2 /( gL) is proportional to the ratio of kinetic and potential energies. The sedimentation Reynolds number Re = Ud1 / ν 0f is the ratio between inertial and viscous forces, and d1 is the diameter of the largest particle. In our application, the size of the largest particle d1~10-6m, the height of the settling vessel L~10-1 m, g~10m2/s, ρ f =103kg/m3, and v0f =10-6 m2/s. Based on the Stokes velocity, we estimate U~10-5m/s. Accordingly, Fr=10-10, Re=10-5, and d1/L=10-5. It is also reasonable to assume that ν 0s 30 min).

6.2 Sandwich Sedimentation Biodetector In the absence of the target analyte A (i.e., [A]0=0), the protein-conjugated reporter particles P cannot bind to the protein-conjugated beads B, and there are only two particle species B and P in solution. Since the beads B settle much faster than the reporter particles P, eventually the beads B accumulate at the reactor’s bottom, leaving the reporter particles in suspension. The presence of a high concentration of reporter particles in suspension indicates the absence of the target analyte. The process is depicted schematically in Fig. 1a.

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

Fig. 7 depicts quantitatively the concentration of the reporter particles P as a function of time and space. Witness that the concentration of the reporter particles is nearly uniform throughout most of the reactor chamber; but it declines sharply next to the bottom, in the region occupied by the settling beads. To better visualize the concentration distribution of the reporter particles P next to the bottom, Fig. 8 depicts the concentration of reporter particles P as a function of time at various x-locations. When the heavier beads B settle, they displace the lighter particles P, leading to a lower concentration of the indicator particles P in the sediment layer. Witness the oscillations in the reporter particle concentration in the lower part of the reactor. These oscillations are caused by the interactions between the downward wave associated with the settling of the beads and the upward wave associated with the movement of the liquid and reporter particles. Eventually these oscillations decay. Next, we investigate the effect of the target analyte concentration on the bead and reporter particles’ distributions. When the target analytes are present, the reporter particles can bind to the beads and settle. Thus, the presence of target analyte is indicated by the depletion of reporter particles in the supernatant. The process is depicted schematically in Fig. 1b. Figs. 9 and 10 depict, respectively, the concentrations of the complex AP and the free reporter particles P as functions of space and time when the initial target analyte concentration [A]0=10 nM. The rate of formation of the complex AP is highest at time t=0, and it decreases as time increases. As in Fig. 4, the Smith effect (Smith, 1966) which causes an excess concentration of reporter particles next to the surface of the reaction chamber (x=0.1 m), is evident in both Figs. 9 and 10. Due to the binding of AP with the beads B to form the complex BAP, Fig. 9 depicts a low concentration of the AP complexes throughout most of the chamber’s volume. In the presence of target analyte, there are few free reporter particles (Fig.10) in the bulk of the solution. The measured signal is proportional to the total concentration of the reporter particles S=[P]+[AP]+[BAP]. Fig. 11 depicts the signal level S in the presence of the target analyte at CES-D-04-00047

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

initial concentration [A]0=10 nM (solid lines) and in the absence of the target analyte (dashed lines) at times 10 minutes (a), 20 minutes (b), and 30 minutes (c). For better visibility, the figure is truncated at S~1.1 nM. Fig. 11 mimics the signal that would have been detected with a scanner. Only very slow changes are observed after 30 minutes, indicating that the signal is nearly fully developed within the first 30 minutes. In the absence of the target analyte (dashed lines), the supernatant’s signal is much higher than in the presence of the target analyte (solid line).

7. Conclusions A mathematical model and numerical scheme for modeling sedimentation bioreactors is proposed. The model allows one to predict the spatial and temporal distributions of each species’ concentration under various conditions. Our model is a fusion of two previously well-studied models: a mathematical model for the sedimentation of particles of various sizes in the absence of biological interactions and a model for biological interactions in the presence of a specified flow field. The predictions of the sedimentation model for the settling of poly-disperse suspensions with particles of various sizes and densities in the absence of biological interactions were compared and favorably agreed with the experimental observations of Smith (1965), Selim et al. (1983), El-Genk et al. (1985), Law et al. (1987), Xue and Sun (2003), and Xue et al. (2003). The predictions of the biological interactions model that accounts for mass transfer when the flow field is apriori known were also compared and favorably agreed with experimental data (Qian and Bau, 2003 and 2004). We were not able to find in the existing literature any experimental data for the sedimentation reactor that we studied here. Nevertheless, the agreement between the model’s predictions and experimental data in the special cases discussed above gives us confidence that the theoretical predictions are reliable. We hope that the model presented here will be useful to designers of sedimentation biodetectors. The simulations can be used to predict reactor performance as well as to assist in the selection of reporter particle and bead concentrations to achieve optimal detection conditions CES-D-04-00047

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

for a specified concentration range of target analytes. Although the numerical simulations cannot substitute for experiments, they can help narrow the experimental parameter space, shorten the development process, and increase the probability of success. The presence of target analytes both in the competitive and sandwich assays is detected by monitoring the concentration of reporter particles in the bulk of the solution. In the case of the sandwich assay, depletion in the reporter particle concentration indicates the presence of target analyte. In contrast, in the case of the competitive assay, the presence of reporter particles in the bulk of the solution indicates the presence of target analyte. The computations indicate the sedimentation process is relatively slow. The sedimentation of the beads can be significantly accelerated by selecting larger diameter beads and/or by using centrifugal, magnetic, and electric fields to increase the settling force. Of course, the sedimentation time must be long enough to allow sufficient time for the biological interactions. The work presented here can be expanded in a number of directions. Better models are needed for the interactions between molecules in solution and particles. The reactor can be modified to act as a flow-through reactor. In that case the unbound target analytes and reporter particles will be free to flow through a membrane while the beads and the bead-target analytereporter particle complexes will remain behind.

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APPENDIX: In this appendix, we provide additional details on the implementation of the numerical scheme. We first describe the procedures used to calculate the flux h nj ±1 . Given the vector W jn (j=1, 3,…, J-1), we construct a piecewise linear interpolation of Wjn at time tn. To this end, we need to determine the slope vector W ′j = (W1′, j , K , W N′ , j ) T (j=1, 3,…, J-1), where

when j = 1 and j = J − 1 ⎧⎪0 Wi′, j = ⎨ , (A1) n n n n n n ⎪⎩MM θ Wi, j − Wi, j −2 , Wi, j +2 − Wi, j −2 / 2,θ Wi, j +2 − Wi, j when j = 3,5,K, J − 3

{(

)(

)

(

)}

i=1,…,N, and ⎧min(a, b, c) ⎪ MM (a, b, c) = ⎨max(a, b, c) ⎪0 ⎩

when a, b, c > 0 when a, b, c < 0

(A2)

otherwise

is the minmod function and θ ∈ (0, 2) . The choice of θ is problem-dependent (Berres et al., 2003). In our simulations, we used θ=1.3. The values of W and the maximal wave speeds at the cell boundaries xj (j=2, 4, …, J2) are, respectively,

1 Wjm = Wjnm1 ± W′j m1 , 2

j = 2,4,..., J − 2

(A3)

and

a nj = max{ρ (J F ( W j− ) ), ρ (J F ( W j+ ) )}, j = 2,4,..., J − 2 ,

(A4)

) ,

) ,

where J F ( W ) = ∂Fi ( W q ∂W j (i, j=1,…,N) is the Jacobian of F( W q ; λi (J F ) is the i-th eigenvalue of the matrix JF; and

ρ (J F ) = max λi (J F ) is the spectral radius of JF. i

Consistent with our order of magnitude analysis, ∂q ∂W j is small compared to the other terms in the Jacobian, and we set ∂q ∂W j ≈ 0 .

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

The half-cell averages of the left ([xj-1, xj]) and the right ([xj, xj+1]) half-cells adjacent to x=xj are denoted, respectively, with subscripts L and R.

-

⎛1 ⎞ W jn, L = W jn−1 + ⎜ λa nj ⎟ W′j −1 , ⎝2 ⎠

j = 2,4,..., J − 2

(A5)

and

-

⎛1 ⎞ W jn, R = W jn+1 − ⎜ λa nj ⎟ W′j +1 , ⎝2 ⎠

j = 2,4,..., J − 2 .

(A6)

The flux slope vector

F′( W jn,c ) = (F1′( W jn,c ),K, FN′ ( W jn,c ) ) ,

c = L, R and j = 3,4,..., J − 3

T

Fi′( W2n,c ) = Fi ′( WJn− 2,c ) = 0,

c = L, R

(A7) (A8)

and

Fi′(Wjn,c ) = MM{α , β ,ϑ}, c = L , R; i = 1,..., N ; and j = 4,6,...,J − 4 .

(A9)

In the above,

α = θ (Fi ( W jn,c , q~ n ( x j )) − Fi ( W jn−2,c , q~ n ( x j −2 )) ) ,

(A10)

β = (Fi ( W jn+ 2,c , q~ n ( x j + 2 )) − Fi ( W jn− 2,c , q~ n ( x j − 2 )) ) / 2 ,

(A11)

ϑ = θ (Fi ( W jn+ 2,c , q~ n ( x j + 2 )) − Fi ( W jn,c , q~ n ( x j )) ) ,

(A12)

The function q~ n ( x) is the approximation of the function q(x, tn) obtained from the solution vector W jn by a quadrature rule applied to (37). For example, l

(

)

(

~ W n + ( x − x )m ~ Wn q~ n ( x) = 2Δx ∑ m 2 j −1 2l 2 l +1

)

j =1

when x ∈ ( x 2l , x 2l + 2 ]

(A13)

We calculate the midpoint values with Taylor series expansions:

W jn,+c1 / 2 = W jn,c −

λ ′ n F ( W j ,c ), c = L, R and j = 2,4,..., J − 2 . 2

(A14)

Next, we define the cell averages at time t=(n+1)Δt:

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ψ

n +1 j

(W =

n j −1

+ Wjn+1 ) 2

(1 − λa )(W′ + n j

j-1

− W′j +1 )

4

1 − n 2a j

⎡ ⎛ n+1 / 2 ~ n+1 / 2 ⎛ a nj Δt ⎞ ⎞ ⎜ ⎜⎜ x j + ⎟⎟ ⎟ − ⎢F⎜ Wj , R , q ⎟ 2 ⎢⎣ ⎝ ⎝ ⎠⎠

⎛ a Δt ⎞ ⎞⎤ ⎛ ⎟⎟ ⎟⎥, j = 2,4,...,J − 2 F⎜⎜ Wjn,+L1 / 2 , q~ n+1 / 2 ⎜⎜ x j − ⎟ 2 ⎝ ⎠ ⎠⎥⎦ ⎝

(A15)

n j

over the non-uniform interval [ x j − a nj Δt , x j + a nj Δt ] and

ψnj+1 = Wjn −

λ(anj+1 − anj−1 )W′j 2

−

⎡ ⎛ n+1/ 2 ~n+1/ 2 ⎛ Δx − anj−1Δt ⎞⎞ ⎜ ⎟⎟⎟ − ⎜ , q x F W − ⎢ j +1, L ⎜ j ⎟ 1 − λ(anj−1 + anj+1 ) ⎣⎢ ⎜⎝ 2 ⎠⎠ ⎝

λ

⎛ Δx − a Δt ⎞⎞⎤ ⎛ ⎟⎟⎟⎥, j = 3,5,..., J − 3 F⎜⎜ Wjn−+11,R/ 2 , q~n+1/ 2 ⎜⎜ x j + ⎟ 2 ⎠⎠⎦⎥ ⎝ ⎝

(A16)

n j +1

over the interval [ x j −1 + a nj−1 Δt , x j +1 − a nj+1 Δt ] . In the above, l

[(

) (

~ W n +1 / 2 + m ~ W n +1 / 2 q~ n +1 / 2 ( x) = Δx ∑ m 2 j −1, L 2 j −1, R j =1

(

)

)]

~ W n +1 / 2 ⎧⎪( x − x 2l )m 2 l +1, L +⎨ n +1 / 2 ~ ~ W n +1 / 2 ⎪⎩Δxm W2l +1, L + ( x − x 2l +1 )m 2 l +1, R

(

)

(

)

when x ∈ [ x 2l , x 2l +1 ),

(A17)

when x ∈ [ x 2l +1 , x 2l + 2 ).

Using both families of the approximate cell averages, we determine the vector of discrete ψ

ψ

= 0 and

2 J

2

j , N

j , 1

derivatives: ψ ′j = ( ′ , K , ′ ) T ( j = 2,4,..., J − 2 ) , where ψ ′ = ψ ′

⎧⎪ ψ in, j+1 −ψ in, j+−11 ψ in, j++11 −ψ in, j+−11 ψ in, j++11 −ψ in, j+1 1 ψ i′, j = MM⎨θ , ,θ n n n n n n n Δx ⎪⎩ 1 + λ a j − a j −2 2 + λ 2a j − a j −2 − a j +2 1 + λ a j − a j +2

(

)

(

)

(

⎫⎪ ⎬ ⎪⎭

)

(A18)

for (i=1,…,N) and (j=4,6,…, J−4). Next, we calculate the desired numerical flux vectors:

Δx ⎞ ⎞ ⎛ Δx ⎞ ⎞⎤ 1⎡ ⎛ ⎛ ⎛ h nj = ⎢F⎜ Wjn,+R1 / 2 , q~ n+1 / 2 ⎜ x j + ⎟ ⎟ + F⎜ Wjn,+L1 / 2 , q~ n+1 / 2 ⎜ x j − ⎟ ⎟⎥ 2⎣ ⎝ 2 ⎠⎠ ⎝ 2 ⎠ ⎠⎦ ⎝ ⎝ (A19) n n n n n ′ ′ a j (Wj +1 − Wj −1 ) a j (1 − λa j )(Wj −1 + Wj +1 ) 2 − + + λΔx(a nj ) ψ′j , j = 2, 4,..., J − 2 2 4 Finally, we outline the procedure to calculate the source term. An overview of various discretization schemes of source terms such as S(W) appearing in (32) is given in Russo (2002). In our application, the source terms are not stiff, and we utilize a fully explicit time CES-D-04-00047

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

discretization. To this end, we replace the formula for calculating the predictor solution values at t=tn+1/2, (A14), with

λ Δt Wjn,+c1 / 2 = Wjn,c − F′ Wjn,c + S Wjn,c , c = L, R, j = 2,4,...,J − 2 , 2 2

(

S nj =

)

(A20)

1 S Wjn++11,L/ 2 + S Wjn−+11,R/ 2 , j = 3,5,...,J − 3 , 2

)]

(A21)

S1n = S W1n ,

( )

(A22)

(

(A23)

[(

)

(

) (

and

)

S nJ −1 = S WJn−1 .

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

Acknowledgments The work described in this paper was supported, in part, by the NIH Grant 1U01 DE 14964-01 and by the DARPA SIMBIOSYS Program (N66001-01-C-8056). We also acknowledge support by the Collaborative Research Center (Sonderforschungsbereich) 404 at the University of Stuttgart.

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References 1. Arastoopour, H., Lin, S.C., Weil, S.A., 1982. Analysis of vertical pneumatic conveying of solids using multiphase flow models. AICHE J., 28, 467–473. 2. Batchelor, G.K., 1982. Sedimentation in a dilute polydisperse system of interacting spheres. Part 1. General theory. J. Fluid Mech., 119, 379–408. 3. Batchelor, G.K., Wen, C.S., 1982. Sedimentation in a dilute polydisperse system of interacting spheres. Part 2. Numerical results. J. Fluid Mech., 124, 495–528. 4. Berres, S., Bürger, R., Karlsen K.H., Tory, E.M., 2003. Strongly degenerate parabolichyperbolic systems modeling polydisperse sedimentation with compression. SIAM J. Appl. Math., 64, 41–80. 5. Berres, S., Bürger, R., Karlsen, K.H., 2004a. Central schemes and systems of conservation laws with discontinuous coefficients modeling gravity separation of polydisperse suspensions. J. Comp. Appl. Math., 164-165, 53–80. 6. Berres, S., Bürger, R., Tory, E.M., 2004b. Applications of polydisperse sedimentation models. Chem. Eng. J., to appear. 7. Berres, S., Bürger, R., Tory, E.M., 2005. On mathematical models and numerical simulation of the fluidization of polydisperse suspensions. Applied Mathematical Modeling, 29, 159193.

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8. Bürger, R., Tory, E.M., 2000. On upper rarefaction waves in batch settling. Powder Technology, 108, 74–87. 9. Bürger, R., Wendland, W.L., Concha, F., 2000a. Model equations for gravitational sedimentation-consolidation processes. Z. Angew. Math. Mech. 80, 79–92. 10. Bürger, R., Concha, F., Tiller, F.M., 2000b. Applications of the phenomenological theory to several published experimental cases of sedimentation processes. Chem. Eng. J. 80, 105-117. 11. Bürger, R., Concha, F., Fjelde, K.K., Karlsen, K.H., 2000c. Numerical simulation of the settling of polydisperse suspensions of spheres. Powder Technology, 113, 30–54. 12. Bürger, R., Wendland, W.L., 2001. Sedimentation and suspension flows: Historical perspective and some recent developments. J. Eng. Math., 41, 101–116. 13. Bürger, R., Fjelde, K.K., Höfler, K., Karlsen, K.H., 2001. Central difference solutions of the kinematic model of settling of polydisperse suspensions and three-dimensional particle-scale simulations. J. Eng. Math., 41, 167–187. 14. Bürger, R., Karlsen, K.H., Tory, E.M., Wendland, W.L., 2002. Model equations and instability regions for the sedimentation of polydisperse suspensions of spheres. Z. Angew. Math. Mech., 82, 699–722. 15. Concha, F., Bürger, R., 2002. A century of research in sedimentation and thickening. KONA Powder and Particle, 20, 38–70.

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16. Davis, R.H., Gecol, H., 1994. Hindered settling function with no empirical parameters for polydisperse suspensions. AICHE J., 40, 570–575. 17. Drew, D.A., Passman, S.L., 1999. Theory of multicomponent fluids, Springer-Verlag, New York, Inc. 18. El-Genk, M.S., Kim, S.-H., and Erickson, D., 1985. Sedimentation of binary mixtures of particles of unequal densities and of different sizes. Chemical Engineering Communications, 36, 99-119. 19. Feleszko, W., Maksymiuk, J., Oracz, G., Golicka, D., Szajewska, H., 2004. The TUBEX (TM) typhoid test detects current Salmonella infections. Journal of Immunological Methods, 285, 137–138. 20. House, D., Wain, J., Ho, V.A., Diep, T.S., Chinh, N.T., Bay, P.V., Vinh, H., Duc, M., Parry, C.M., Dougan, G., White, N.J., Hien, T.T., Farrar, J.J., 2001. Serology of typhoid fever in an area of endemicity and its relevance to diagnosis. Journal of Clinical Microbiology, 39, 1002–1007. 21. Kurganov, A., Tadmor, E., 2000. New high resolution central schemes for nonlinear conservation laws and convection-diffusion equations. Journal of Computational Physics, 160, 241–282.

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22. Law, H.S., Masliyah, J.H., MacTaggart, R.S., Nandakumar, K., 1987. Gravity separation of bidisperse suspensions: light and heavy particle species, Chemical Engineering Science, 42, 1527–1538. 23. Lim, P.L., 1990. A one-step 2-particle latex immunoassay for the detection of salmonellatyphi endotoxin. Journal of Immunological Methods, 135, 257–261. 24. Lim, P.L., Ko, K.H., 1990. A tube latex test based on color separation for the detection of IgM antibodies to either one of 2 different microorganisms. Journal of Immunological Methods, 135, 9–14 25. Lim, P.L., Tam, F.C.H., Cheong, W.M., Jegathesan, M., 1998. One-step 2-minute test to detect typhoid-specific antibodies based on particle separation in tubes. Journal of Clinical Microbiology, 36, 2271–2278. 26. Lockett, M.J., Al-Habbooby, H.M., 1973. Differential settling by size of two particle species in a liquid. Transactions of the Institution of Chemical Engineers, 51, 281–292. 27. Lockett, M.J., Bassoon, K.S., 1979. Sedimentation of binary particle mixtures. Power Technol., 24, 1–7. 28. Masliyah, J.H., 1979. Hindered settling in a multi-species particle system. Chem. Eng. Sci., 34, 1166–1168. 29. Mirza, S., Richardson, J.F., 1979. Sedimentation of suspensions of particles of two or more sizes. Chem. Eng. Sci., 34, 447–454.

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30. Nakamura, K., Capes, C.E., 1976. Vertical pneumatic conveying of binary particle mixtures. In: Fluidization Technology, Vol. 2, Keairns, D.L. (Eds.), Hemisphere Publishing, Washington, DC, 159–184. 31. Oracz, G.., Feleszko, W., Golicka, D., Maksymiuk, J., Klonowska, A., Szajewska, H., 2003. Rapid diagnosis of acute Salmonella gastrointestinal infection. Clinical Infectious Diseases, 36, 112–115. 32. Qian, S., Bau, H.H., 2003. A mathematical model of lateral flow bio-reactions applied to sandwich assays. Analytical Biochemistry, 322, 89–98. 33. Qian, S., Bau, H.H., 2004. Analysis of lateral flow bio-detector: competitive assay. Analytical Biochemistry, 326, 211–224. 34. Richardson, J. F., Zaki, W.N., 1954. Sedimentation and fluidization: Part I. Trans. Inst. Chem. Engrs. (London), 32, 35–53. 35. Russo, G., 2002. Central Schemes and Systems of Balance Laws, In: Meister, A., and Struckmeier, J., (Eds.), Hyperbolic Partial Differential Equations, Theory, Numerics and Applications, Vieweg, Braunschweig, 59–114. 36. Schuck, P., 1998. Sedimentation analysis of noninteracting and self-associating solutes using numerical solutions to the Lamm equation. Biophysical Journal, 75, 1503–1521. 37. Schuck, P., 2004a. A model for sedimentation in inhomogeneous media. I. Dynamic density gradients from sedimenting co-solutes. Biophysical Chemistry, 108, 187–200.

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38. Schuck, P., 2004b. A model for sedimentation in inhomogeneous media. II. Compressibility of aqueous and organic solvents. Biophysical Chemistry, 108, 201–214. 39. Selim, M.S., Kothari, A.C., Turian, R.M., 1983. Sedimentation of multisized particles in concentrated suspensions. AICHE J., 29, 1029–1038. 40. Sharma, R.V., Edwards, R.T., Beckett, R., 1993. Physical characterization and quantification of bacteria by sedimentation field-flow fractionation. Appl. Environ. Microbiol., 59, 1864–1875. 41. Shih, Y.T., Gidaspow, D., Wasan, D.T., 1987. Hydrodynamics of sedimentation of multisized particles, Powder Technology, 50(1987), 201–215 42. Smith, T.N., 1965. The differential sedimentation of particles of two different species. Transactions of the Institution of Chemical Engineers, 43, T69–T73. 43. Smith, T.N., 1966. The sedimentation of particles having a dispersion of sizes. Transactions of the Institution of Chemical Engineers, 44, 153–157. 44. Stafford, W.F., Braswell, E.H., 2004. Sedimentation velocity, multi-speed method for analyzing polydisperse solutions. Biophysical Chemistry, 108, 273–279. 45. Tam, F.C.H., Lim, P.L., 2003. The TUBEX (TM) typhoid test based on particleinhibition immunoassay detects IgM but not IgG anti-O9 antibodies. Journal of Immunological Methods, 282, 83–91.

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46. Tory, E.M., Ford, R.A., 2004. Simulation of sedimentation of bidisperse suspensions. Int. J. Mineral Process, 73, 119–130. 47. Xue, B., Sun, Y., 2003. Modeling of sedimentation of polydisperse spherical beads with a broad size distribution. Chem. Eng. Sci., 58, 1531–1543. 48. Xue, B., Tong, X., Sun, Y., 2003. Polydisperse model for the hydrodynamics of expanded bed adsorption systems. AIChE J., 49, 2510 – 2518. 49. Zeidan, A., Rohani, S., Bassi, A., 2004. Dynamic and steady-state sedimentation of polydisperse suspension and prediction of outlets particle-size distribution. Chemical Engineering Science, 59, 2619–2632.

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List of Captions 1. A schematic diagram of the sedimentation biodetector operating in a sandwich assay format in the absence (A) and presence (B) of target analytes. I and II denote, respectively, initial and final conditions. The symbols

,

, and

represent, respectively, the bead (B), the indicator

particle (P), and the target analyte (A). 2. A schematic diagram of the sedimentation bio-detector operating in a competitive assay format in the absence (A) and presence (B) of target analytes. I and II denote, respectively, initial and final conditions. The symbols

,

, and

represent, respectively, the bead (B),

the indicator particle (P), and the target analyte (A). 3. The concentration of the complex BP in the competitive sedimentation biodetector as a function of space and time. [A]0=0 nM, [B]0=10 nM, and [P]0=1 nM. 4. The concentration of the reporter particles P in the competitive sedimentation biodetector as a function of space and time. [A]0=0, [B]0=10 nM, and [P]0=1 nM. 5. The concentration of the reporter particles P in the competitive sedimentation biodetector as a function of space and time. [A]0=10 nM, [B]0=10 nM, and [P]0=1 nM. 6. The signal S=[P]+[BP] as a function of x at 10 minutes (a), 20 minutes (b), and 30 minutes (c) in the presence [A]0=10 nM (solid line) and in the absence of target analyte [A]0=0 (dashed line) during the competitive sedimentation biodetector. 7. The concentration of the reporter particles P in the sandwich sedimentation biodetector as a function of space and time. [A]0=0, [B]0=10 nM, and [P]0=1 nM. 8. The concentration of the reporter particles P in the sandwich sedimentation biodetector as a function of time at various locations near the bottom of the reactor. [A]0=0, [B]0=10 nM, and

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

[P]0=1 nM. 9. The concentration of the complex AP in the sandwich sedimentation biodetector as a function of space and time. [A]0=10 nM, [B]0=10 nM, and [P]0=1 nM. 10. The concentration of the reporter particles P in the sandwich sedimentation biodetector as a function of space and time. [A]0=10 nM, [B]0=10 nM, and [P]0=1 nM. 11. The signal S=[P]+[BP]+[BAP] as a function of x at 10 minutes (a), 20 minutes (b), and 30 minutes (c) in the presence of target analyte [A]0=10 nM (solid line) and in the absence of target analyte [A]0=0 nM (dashed line) during the sandwich sedimentation reactor.

LIST OF TABLES 1. The material properties of the particle and fluid species involved in the competitive and sandwich formats 2. The interaction rate constants of the reactions involved in the competitive and sandwich formats

CES-D-04-00047

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

(AI)

(AII)

(BI)

(BII)

Fig.1: A schematic diagram of the sedimentation biodetector operating in a sandwich assay format in the absence (A) and presence (B) of target analytes. I and II denote, respectively, initial and final conditions. The symbols

,

, and

represent, respectively, the bead

(B), the indicator particle (P), and the target analyte (A).

CES-D-04-00047

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

(AI)

(AII)

(BI)

(BII)

Fig.2: A schematic diagram of the sedimentation bio-detector operating in a competitive assay format in the absence (A) and presence (B) of target analytes. I and II denote, respectively, initial and final conditions. The symbols

,

, and

represent, respectively,

the bead (B), the indicator particle (P), and the target analyte (A).

CES-D-04-00047

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

Fig.3: The concentration of the complex BP in the competitive sedimentation biodetector as a function of space and time. [A]0=0 nM, [B]0=10 nM, and [P]0=1 nM.

CES-D-04-00047

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

Fig.4: The concentration of the reporter particles P in the competitive sedimentation biodetector as a function of space and time. [A]0=0, [B]0=10 nM, and [P]0=1 nM.

CES-D-04-00047

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

Fig.5: The concentration of the reporter particles P in the competitive sedimentation biodetector as a function of space and time. [A]0=10 nM, [B]0=10 nM, and [P]0=1 nM.

CES-D-04-00047

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

Fig.6: The signal S=[P]+[BP] as a function of x at 10 minutes (a), 20 minutes (b), and 30 minutes (c) in the presence [A]0=10 nM (solid line) and in the absence of target analyte [A]0=0 (dashed line) during the competitive sedimentation biodetector.

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

Fig.7: The concentration of the reporter particles P in the sandwich sedimentation biodetector as a function of space and time. [A]0=0, [B]0=10 nM, and [P]0=1 nM.

CES-D-04-00047

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

Fig.8: The concentration of the reporter particles P in the sandwich sedimentation biodetector as a function of time at various locations near the bottom of the reactor. [A]0=0, [B]0=10 nM, and [P]0=1 nM.

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

Fig.9: The concentration of the complex AP in the sandwich sedimentation biodetector as a function of space and time. [A]0=10 nM, [B]0=10 nM, and [P]0=1 nM.

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

Fig.10: The concentration of the reporter particles P in the sandwich sedimentation biodetector as a function of space and time. [A]0=10 nM, [B]0=10 nM, and [P]0=1 nM.

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

Fig.11: The signal S=[P]+[BP]+[BAP] as a function of x at 10 minutes (a), 20 minutes (b), and 30 minutes (c) in the presence of target analyte [A]0=10 nM (solid line) and in the absence of target analyte [A]0=0 nM (dashed line) during the sandwich sedimentation reactor.

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

Table 1: The material properties of the particle and fluid species involved in the competitive and sandwich formats Species

di [10-6 m]

ρi [kg/m3]

MWi [kg/mol]

1

Protein-conjugated settling bead B

5.0

5300

1.0×1010

2

Protein-conjugated reporter particle P

0.1

1300

1.07×1011

3

Complex AP

0.1

1300

1.07×1011

4

Complex BP

5.0

5300

1.17×1011

5

Complex BA

5.0

5300

1.0×1010

6

Complex BAP

5.0

5300

1.17×1011

7

Target analyte A

N/A

N/A

150

8

Buffer solution

N/A

1000.

N/A

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Qian, S. Burger, R., and Bau, H., H., 2005, Analysis of Sedimentation Biodetectors, Chemical Engineering Science, 60, 2585 – 2598

Table 2: The interaction rate constants of the reactions involved in the competitive and sandwich formats

Interaction

ka (1/Ms)

kd (1/s)

B+A=BA

107

10-3

Competitive and sandwich assays

P+A=AP

106

10-3

Sandwich assay

B+P=BP

106

10-3

Competitive assay

B+AP=BAP

107

10-3

Sandwich assay

BA+P=BAP

106

10-3

Sandwich assay

CES-D-04-00047

Comments

49